Problem 133 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}-9=(x-3)^{2}\) for any real number \(x\)

Step-by-Step Solution

Answer

The statement \(x^{2}-9=(x-3)^{2}\) for any real number \(x\) is false. The corrected, true statement would be \(x^{2}-9=(x+3)(x-3)\).

Step 1: Simplify the Left Side

The left side of the statement is \(x^{2}-9\). The number 9 is a perfect square, so this can be written as a difference of squares: \(x^{2}-3^{2}\).

Step 2: Simplify the Right Side

The right side of the statement is \((x-3)^{2}\). The square of a binomial \(a - b\) is \(a^{2} - 2ab + b^{2}\). Following this rule, \((x-3)^{2}\) can be written as \(x^{2} - 2*3*x + 3^{2}\), or \(x^{2} - 6x + 9\).

Step 3: Compare the Results

Now that both sides are simplified, we have \(x^{2} - 9\) on the left side and \(x^{2} - 6x + 9\) on the right side. Their expressions are not the same for all real number \(x\), indicating that the statement is false.

Step 4: Make a True Statement

The statement can be made true by making each side equal to each other. By comparing with each side, a true statement would be \(x^{2}-9=(x+3)(x-3)\)

Key Concepts

Difference of Squares

The concept of the difference of squares is a key topic in algebra. When you see an expression like \(x^2 - 9\), you can recognize it as a difference of squares. This is because 9 is a perfect square, specifically \(3^2\). The difference of squares formula states that \(a^2 - b^2 = (a-b)(a+b)\). This formula is handy for factoring such expressions easily.
In our problem, \(x^2 - 9\) can be expressed as \((x-3)(x+3)\).

• "\(a\)" in the formula is "\(x\)".
• "\(b\)" is "3" because \(9 = 3^2\).

Using this rule, we can break apart polynomials that take the form of \(x^2 - m^2\) and rewrite them as products of two binomials \((x-m)(x+m)\). This is a very effective technique in simplifying and solving algebraic equations.

Binomials

A binomial is simply a mathematical expression that contains two terms. In this context, the expression \((x-3)^2\) is called a binomial square. Calculating a square of a binomial involves using the pattern: \((a-b)^2 = a^2 - 2ab + b^2\).
So for \((x-3)^2\):

• \(a\) is \(x\).
• \(b\) is \(3\).
• Expanding gives: \(x^2 - 2(x)(3) + 3^2\) which simplifies to \(x^2 - 6x + 9\).

Understanding binomials and binomial expansions is crucial because they often appear in polynomial multiplication and simplification tasks. Recognizing patterns like these makes algebra problems far more approachable and less daunting.

False Mathematical Statements

Mathematical statements can sometimes be misleading or incorrect. As with the exercise, it is necessary to verify the expressions on each side of an equation. When the equation claims that \(x^2 - 9 = (x-3)^2\), verifying by simplifying can reveal a falsehood.

• The left side simplifies to \(x^2 - 3^2\), which factors to \((x-3)(x+3)\).
• The right side simplifies to \(x^2 - 6x + 9\) after expanding the binomial.

By comparing these, it's evident that \(x^2 - 9\) is not equal to \(x^2 - 6x + 9\). This discrepancy invalidates the original claim. To establish a true statement, you must adjust one or both sides so they match appropriately. Here, rewriting the left side in its factored form accurately fixes the statement by equaling both sides as \(x^2 - 9 = (x+3)(x-3)\), thus correcting any errors.